John Smolin coined the phrase "Going to the Church of the Larger Hilbert Space" for the dilation constructions of channels and states, which not only provide a neat characterization of the set of permissible quantum operations but are also a most useful tool in quantum information science.
According to Stinespring's dilation theorem, every completely positive and trace-preserving map, or channel, can be built from the basic operations of (1) tensoring with a second system in a specified state, (2) unitary transformation, and (3) reduction to a subsystem. Thus, any quantum operation can be thought of as arising from a unitary evolution on a larger (dilated) system. The auxiliary system to which one has to couple the given one is usually called the ancilla of the channel. Stinespring's representation comes with a bound on the dimension of the ancilla system, and is unique up to unitary equivalence.
Stinespring's dilation theorem
We present Stinespring's theorem in a version adapted to completely positive and trace-preserving maps between finite-dimensional quantum systems. For simplicity, we assume that the input and output systems coincide. The theorem applies more generally to completely positive (not necessarily trace-preserving) maps between C * − algebras.
- Stinespring's dilation: Let be a completely positive and trace-preserving map between states on a finite-dimensional Hilbert space H. Then there exists a Hilbert space and a unitary operation U on such that
- for all , where denotes the partial trace on the system.
- The ancilla space can be chosen such that . This representation is unique up to unitary equivalence.
It is sometimes useful not to go to a larger Hilbert space, but to work with operators between the input and output Hilbert spaces of the channel itself. Such a representation can be immediately obtained from Stinespring's theorem: We introduce a basis of the ancilla space and define the Kraus operators tk in terms of Stinespring's unitary U as
The Stinespring representation then becomes the operator-sum decomposition or Kraus decomposition of the quantum channel T:
- Kraus decomposition: Every completely positive and trace-preserving map can be given the form
- for all . The Kraus operators satisfy the completeness relation .
Purification of quantum states
where is a pure state on the combined system . In other words, every mixed state can be thought of as arising from a pure state on a larger Hilbert space. This special version of Stinespring's theorem is usually called the GNS construction of quantum states, after Gelfand and Naimark, and Segal.
For a given mixed state with spectral decomposition , such a purification is given by the state
References and further reading
- M. A. Nielsen, I. L. Chuang: Quantum Computation and Quantum Information; Cambridge University Press, Cambridge 2000
- K. Kraus: States, Effects, and Operations; Springer, Berlin 1983
- E. B. Davies: Quantum Theory of Open Systems; Academic Press, London 1976
- V. Paulsen: Completely Bounded Maps and Operator Algebras; Cambridge University Press, Cambridge 2002
- M. Keyl: Fundamentals of Quantum Information Theory; Phys. Rep. 369 (2002) 431-548; quant-ph/0202122
- W. F. Stinespring: Positive Functions on C * − algebras; Proc. Amer. Math. Soc. 6 (1955) 211
- I. M. Gelfand, M. A. Naimark: On the Imbedding of Normed Rings into the Ring of Operators in Hilbert space; Mat. Sb. 12 (1943) 197
- I. E. Segal: Irreducible Representations of Operator Algebras; Bull. Math. Soc. 61 (1947) 69